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StirlingS2






Mathematica Notation

Traditional Notation









Integer Functions > StirlingS2[n,m] > Specific values > Specialized values > For fixed n





http://functions.wolfram.com/04.15.03.0034.01









  


  










Input Form





StirlingS2[n, 9] == (1/362880) (9 - 9 2^(2 + n) - 63 2^(1 + 2 n) + 28 3^(1 + n) - 7 2^(2 + n) 3^(1 + n) + 126 5^n + 36 7^n - 9 8^n + 9^n - KroneckerDelta[n]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mn> 9 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 362880 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> + </mo> <mrow> <mn> 126 </mn> <mo> &#8290; </mo> <msup> <mn> 5 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <mn> 36 </mn> <mo> &#8290; </mo> <msup> <mn> 7 </mn> <mi> n </mi> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mn> 8 </mn> <mi> n </mi> </msup> </mrow> <mo> + </mo> <msup> <mn> 9 </mn> <mi> n </mi> </msup> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 63 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> StirlingS2 </ci> <ci> n </ci> <cn type='integer'> 9 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 362880 </cn> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 126 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 36 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <cn type='integer'> 8 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 9 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 63 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["n_", ",", "9"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["9", "-", RowBox[List["9", " ", SuperscriptBox["2", RowBox[List["2", "+", "n"]]]]], "-", RowBox[List["63", " ", SuperscriptBox["2", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]]]], "+", RowBox[List["28", " ", SuperscriptBox["3", RowBox[List["1", "+", "n"]]]]], "-", RowBox[List["7", " ", SuperscriptBox["2", RowBox[List["2", "+", "n"]]], " ", SuperscriptBox["3", RowBox[List["1", "+", "n"]]]]], "+", RowBox[List["126", " ", SuperscriptBox["5", "n"]]], "+", RowBox[List["36", " ", SuperscriptBox["7", "n"]]], "-", RowBox[List["9", " ", SuperscriptBox["8", "n"]]], "+", SuperscriptBox["9", "n"], "-", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "362880"], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02